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302A final exam review

Prepare for your math final exam with this comprehensive review covering key topics from textbooks, explorations, and class notes. Includes practice problems and common mistakes to avoid.

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302A final exam review

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  1. 302A final exam review

  2. What is on the test? • From book: 1.2, 1.3, 1.4, 1.7; 2.3; 3.1, 3.2, 3.3, 3.4; 4.2, 4.3; 5.2, 5.3, 5.4; 6.1, 6.2 • From Explorations: 1.1, 1.4, 1.7; 2.8, 2.9; 3.1; 3.13, 3.15, 3.19, 3.20, 4.2, 4.3, 5.8, 5.9, 5.10, 5.12, 5.13, 5.15, 5.16, 6.3, 6.5, • From Class Notes: Describe the strategies used by the students--don’t need to know the names.

  3. Chapter 1 • A factory makes 3-legged stools and 4-legged tables. This month, the factory used 100 legs and built 3 more stools than tables. How many stools did the factory make? • 16 stools, 13 tables

  4. Chapter 1 • Fred Flintstone always says“YABBADABBADO.” If he writes this phrase over and over, what will the 246th letter be? • D

  5. Chapter 2 • Explain why 32 in base 5 is not the same as 32 in base 6. • 32 in base 5 means 3 fives and 2 ones, which is 17 in base 10. • 32 in base 6 means 3 sixes and 2 ones, which is 20 in base 10. So, 32 in base 5 is smaller than 32 in base 6.

  6. Chapter 2 • Why is it wrong to say 37 in base 5? • In base 5, there are only the digits 0, 1, 2, 3, and 4. 7 in base 5 is written 12.

  7. Chapter 2 • What error is the student making? “Three hundred fifty seven is written 300507.” • The student does not understand that the value of the digit is found in the place: 300507 is actually 3 hundred-thousands plus 5 hundreds and 7 ones. Three hundred fifty seven is written 357--3 hundreds plus 5 tens plus 7 ones.

  8. Chapter 3 • List some common mistakes that children make in addition. • Do not line up place values. • Do not regroup properly. • Do not account for 0s as place holders.

  9. Chapter 3 • Is this student correct? Explain. • “347 + 59: add one to each number and get 348 + 60 = 408.” • No: 347 + 59 is the same as 346 + 59 because 346 + 1 + 60 - 1 = 346 + 60 + 1 - 1, and 1 - 1 = 0. The answer is 406.

  10. Chapter 3 • Is this student correct? • “497 - 39 = 497 - 40 - 1 = 457 - 1 = 456.” • No, the student is not correct because 497 - 39 = (497) - (40 - 1) = (497) - 40 + 1 = 458. An easier way to think about this is 499 - 39 = 460, and then subtract the 2 from 499, to get 458.

  11. Chapter 3 • Is this student correct? • “390 - 27 is the same as 300 - 0 + 90 - 20 + 0 - 7. So, 300 + 70 + -7 = 370 + -7 = 363.” • Yes, this student is correct. This is analogous to 390 = 380 + 10 = 27; 300 - 0 + 80 - 20 + 10 - 7 = 300 + 60 + 3. Note: to avoid this negative situation, we regroup.

  12. Chapter 3 • Multiply 39 • 12 using at least 5 different strategies. • Lattice Multiplication • Rectangular Array • Egyptian Duplation • Lightning-Cross • 39 • 10 + 39 • 2 • 40 • 12 - 1 • 12 • 30 • 10 + 9 • 10 + 30 • 2 + 9 • 2 = (30 + 9)(10 + 2)

  13. Chapter 3 • Divide 259 ÷ 15 using at least 5 different strategies. • Scaffold • Repeated subtraction • Repeated addition • Use a benchmark • Partition (Thomas’ strategy)

  14. Chapter 3 • Models for addition: • Put together, increase by, missing addend • Models for subtraction: • Take away, compare, missing addend • Models for multiplication: • Area, Cartesian Product, Repeated addition, measurement, missing factor • Models for division: • Partition, Repeated subtraction, missing factor

  15. • • • • • • Chapter 4 • An odd number: • An even number:

  16. Chapter 4 • Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … 2 factors • ONE IS NOT PRIME. • Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, … at least 3 factors • Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, … an odd number of factors

  17. Chapter 4 • Prime factorization: many ways to get the factorization, but only one prime factorization for any number. • Find the prime factorization of 84. • 2 • 2 • 3 • 7, or 22 • 3 • 7

  18. Chapter 4 • Greatest Common Factor: The greatest number that can divide evenly into a set of numbers. • The GCF of 50 and 75 is 25. • You try: Find the GCF of 60, 80, and 200. • 20: 60 = 20 • 3, 80 = 20 • 4, 200 = 20 • 10.

  19. Chapter 4 • The Least Common Multiple is the smallest number that is divisible by a set of numbers. • The LCM of 50 and 75 is 150. • You try: Find the LCM of 60, 80, and 200. • 1200: 60 • 20 = 1200, 80 • 15 = 1200, 200 • 6 = 1200.

  20. Chapter 4 • What is the largest square that can be used to fill a 6 x 10 rectangle. • 2 x 2: You can draw it to see why. (Which is involved here, GCF or LCM?)

  21. Chapter 5 • Fractions models:Part of a wholeRatioOperatorQuotient • Make up a real-world problem for each model above for 6/10.

  22. Chapter 5 • Name the model for each situation of 5/6. • I have 5 sodas for 6 people--how much does each person get? • Out of 6 grades, 5 were As. • I had 36 gumballs, but I lost 6 of them. What fraction describes what is left? • In a room of students, 50 wore glasses and 10 did not wear glasses.

  23. Chapter 5 • There are three ways to represent a fraction using a part of a whole model:part-wholediscrete,number line (measurement) • Represent 5/8 and 11/8 using each of the pictorial models above.

  24. Chapter 5 • Errors in comparing fractions: 2/6 > 1/2 • Look at the numerators: 2 > 1 • Two pieces is more than one piece. • Look at the denominators: 6 > 2 • We need 6 to make a whole rather than 2. • There are more pieces not shaded than shaded. • If we look at what is not shaded, then there are more unshaded pieces. The pieces are smaller in sixths than in halves.

  25. Chapter 5 • Appropriate ways to compare fractions: • Rewrite decimal equivalents. • Rewrite fractions with common denominators. • Place fractions on the number line. • Sketch parts of a whole, with the same size whole

  26. Chapter 5 • More ways to compare fractions: • Compare to a benchmark, like 1/2 or 3/4. • Same numerators: a/b > a/(b + 1) 2/3 > 2/4 • Same denominators: (a + 1)/b > a/b 5/7 > 4/7 • Look at the part that is not shaded: 5/9 < 8/12 because 4 out of 9 parts are not shaded compared with 4 out of 12 parts not shaded.

  27. Chapter 5 • Compare these fractions without using decimals or common denominators. • 37/81 and 51/90 • 691/4 and 791/7 • 200/213 and 199/214 • 7/19 and 14/39

  28. Chapter 5 • Remember how to compute with fractions. Explain the error: • 2/5 + 5/8 = 7/13 • 3 4/7 + 9/14 = 3 13/14 • 2 7/8 + 5 4/8 = 7 11/8 = 8 1/8 • 5 4/6 + 5/6 = 5 9/6 = 5 1/2

  29. Chapter 5 • Explain the error: • 3 - 4/5 = 2 4/5 • 5 - 2 1/7 = 3 6/7 • 3 7/8 - 2 1/4 = 1 6/4 = 2 1/2 • 9 1/8 - 7 3/4 = 9 2/8 - 7 6/8 = 8 12/8 - 7 6/8 = 1 4/8 = 1 1/2

  30. Chapter 5 • Explain the error: • 3/7 • 4/9 = 7/16 • 2 1/4 • 3 1/2 = 6 1/8 • 7/12 • 4/5 = 35/48 • 4/7 • 3/5 = 20/35 • 21/35 = 420/1225 = 84/245 = 12/35

  31. Chapter 5 • Explain the error: • 3/5 ÷ 4/5 = 4/3 • 12 1/4 ÷ 6 1/2 = 2 1/2

  32. Chapter 5 • Decimals: • Name a fraction and a decimal that is closer to 4/9 than 5/11. • Explain what is wrong: • 3.45 ÷ .05 = 0.69

  33. Chapter 5 • True or false? Explain. • 3.69/47 = 369/470 • 5.02/30.04 = 502/3004

  34. Chapter 5 • Order these decimals: • 3.95, 4.977, 3.957, 4.697, 3.097 • Round 4.976 to the nearest tenth. Explain in words, or use a picture.

  35. Chapter 6 • An employee making $24,000 was given a bonus of $1000. What percent of his salary was his bonus? • 1000/24,000 = x/100 • 100,000 = 24,000x x ≈ 4.17%

  36. Chapter 6 • Which is faster? • 11 miles in 16 minutes or 24 miles in 39 minutes? Explain.

  37. Chapter 6 • Ryan bought 45 cups for $3.15. “0.07! That’s a great rate!” • What rate does 0.07 represent? • Describe this situation with a different rate--and state what this different rate represents.

  38. Chapter 6 • Which ratio is not equivalent to the others? • (a) 42 : 49 • (b) 12 : 21 • (c) 50.4 : 58.8 • (d) 294 : 357

  39. Chapter 6 • Write each rational number as a decimal and a percent. • 3 • 4/5 • 1/11 • 2 1/3

  40. Chapter 6 • Write each decimal as a fraction in simplest form and a percent. • 4.9 • 3.005 • 0.073

  41. Chapter 6 • Write each percent as a fraction and a decimal. • 48% • 39.8% • 2 1/2% • 0.841%

  42. Chapter 6 • A car travels 60 mph, and a plane travels 15 miles per minute. How far does the car travel while the plane travels 600 miles? • (Hint: you can set up one proportion, two proportions, or skip the proportions entirely!) • Answer is the car travels 40 miles--the car travels 1 miles for each 15 miles the plane travels. 1/15 = x/600.

  43. Chapter 6 • DO NOT set up a proportion and solve: use estimation instead. • (a) Find 9% of 360. • (b) Find 5% of 297. • (c) Find 400% of 35. • (d) Find 45% of 784.

  44. Chapter 6 • DO NOT set up a proportion and solve: use estimation instead. • (e) What percent of 80 is 39?(f) What percent of 120 is 31?(g) 27 is what percent of 36?(h) 87 is 20% of what number? • Now, go back and set up proportions to find the exact values of (a) - (h). Were you close?

  45. Chapter 6 • Iga Tahavit has 150 mg of fools’ gold. Find the new amount if: • She loses 30%? • She increases her amount by 90%? • She decreases her amount by 40%?

  46. Percent & Proportion Questions • In Giant World, a giant tube of toothpaste holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost? • One gallon is 128 ounces. Ounces = 4.6 = 128 Dollars $2.49 x4.6x = 128 • 2.49 About $69.29

  47. Estimate • In Giant World, a giant tube of toothpaste holds one gallon. If a normal tube of toothpaste holds 4.6 ounces and costs $2.49, how much should the giant tube cost? • If we round, we can think: 4 ounces is about $2.50. Since we want to know how much 128 ounces is, think: 4 • 32 = 128, so $2.50 times 32 is $80. (or, $2.50 • 30 = $75)

  48. Try this one • The admissions department currently accepts students at a 7 : 3 male/female ratio. If they have about 1000 students in the class, how many more females would they need to reduce the ratio to 2 : 1? • Currently: 7x + 3x = 1000, so x = 100; 700 males and 300 females. They want 2y + 1y = 1000, so y = 333; 666 males and 333 females. They need to accept 333 - 300 = 33 more females to achieve this ratio.

  49. Try this one • Lee’s gross pay is $1840 per paycheck, but $370 is deducted. Her take-home pay is what percent of her gross pay? • Part = percent = 370 = xWhole 100 1840 100 • 370 • 100 = 1840x; About 20% is taken out, so about 80% for take-home pay. • Could also do: 1840 - 370 = 1470: 1470 = x 1840 100

  50. Last one • Estimate in your head: • 16% of 450 • 10% of 450 = 45; 5% = 22.5, about 67.5 OR 10% of 450 = 45; 1% of 450 = 4.5, or about 5; 6 • 1% = 6 • 5 = 30; 30 + 45 = 75. • 123 is approximately what percent of 185? • Approximate: 120 is approximately what percent of 200; 120/200 = 60/100, so about 60%.

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